# On finite volume discretization of infiltration dynamics in tumor growth models

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## Abstract

We address numerical challenges in solving hyperbolic free boundary problems described by spherically symmetric conservation laws that arise in the modeling of tumor growth due to immune cell infiltrations. In this work, we normalize the radial coordinate to transform the free boundary problem to a fixed boundary one, and utilize finite volume methods to discretize the resulting equations. We show that the conventional finite volume methods fail to preserve constant solutions and the incompressibility condition, and they typically lead to inaccurate, if not wrong, solutions even for very simple tests. These issues are addressed in a new finite volume framework with segregated flux computations that satisfy sufficient conditions for ensuring the so-called totality conservation law and the geometric conservation law. We focus on describing the new framework by enhancing the conventional first-order upwind scheme, and the methodology extends to second-order ones. The numerical performance is assessed by various benchmark tests to show that the enhanced method is able to preserve the incompressibility constraint and produce much more accurate solutions than the conventional one.

## Keywords

Finite volume methods Cell incompressibility Free boundary problems Patlak-Keller-Segel system Tumor growth modeling## Mathematics Subject Classification (2010)

65M08 35R35 35Q92## Preview

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## Notes

### Acknowledgments

X. Zeng would like to thank University of Texas at El Paso for the start up support.

### Funding information

J. P. Tian would like to thank the National Science Foundation of US for the support in mathematical modeling under the grant no. DMS-1446139 and the National Institutes of Health of US for the support in cancer research under the grant no. U54CA132383.

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